Midpoint Sum Examples

Use a midpoint sum with 2 sub-intervals to estimate the area between the function f(x) = x² + 1 and the x-axis on the interval [0, 4].

We divide the interval into two sub-intervals [0, 2] and [2, 4], each of width 2:

On sub-interval [0, 2] we go to the midpoint of [0, 2], which is x = 1.

The value of the function at this midpoint gives the height of the rectangle, so the height of the rectangle is f(1) = 2:

The area of this rectangle is

height ⋅ width = 2(2) = 4.

On sub-interval [2, 4] we go to the midpoint of [2, 4], which is x = 3.

The value of the function at this midpoint gives the height of the rectangle, so the height of the rectangle is f(3) = 10:

The area of this rectangle is

height ⋅ width = 10(2) = 20.

We estimate that the area between f and the x-axis on [0, 4] is

4 + 20 = 24.

Partial values of the function g are given in the table below.

• Use a midpoint sum with three sub-intervals to estimate the area between the graph of g and the x-axis on [0, 12].

• Could we use this table to take a midpoint sum with 4 equal sub-intervals? Why or why not?

• Could we use this table to take a midpoint sum with 6 equal sub-intervals? Why or why not?

• Dividing the interval [0, 12] into 3 equal sub-intervals gives us sub-intervals of width 4. We find the function values we need by looking them up in the table.

Sub-interval [0, 4]:

The midpoint of this sub-interval is 2. This means the height of the rectangle is g(2) = 5.

Since the width is 4, the area of this rectangle is

5(4) = 20.

Sub-interval [4, 8]:

The midpoint of this sub-interval is 6. This means the height of the rectangle is g(6) = 11.

Since the width is 4, the area of this rectangle is

11(4) = 44.

Sub-interval [8, 12]:

The midpoint of this sub-interval is 10. This means the height of the rectangle is g(10) = 25.

Since the width is 4, the area of this rectangle is

25(4) = 100.

Adding the areas of these rectangles, we estimate the area between g and the x-axis on [0, 12] to be

20 + 44 + 100 = 164.

• If we divide the sub-interval [0, 12] into 4 equal sub-intervals we get the sub-intervals

[0, 3], [3, 6], [6, 9], and [9, 12].

The midpoints of these sub-intervals are, respectively,

1.5, 4.5, 7.5, and 10.5.

The table doesn't tell us what g is at those places. We can't use the table to find a midpoint sum with 4 equal sub-intervals.

• If we divide the sub-interval [0, 12] into 6 equal sub-intervals we get the sub-intervals

[0, 2], [2, 4], [4, 6], [6, 8], [8, 10], [10, 12].

The midpoints of these sub-intervals are, respectively,

1, 3, 5, 7, 9, 11.

Since the table does tell us what g is at those midpoints, yes, we could use this table to take a midpoint sum with 6 equal sub-intervals.

A graph of the function h is shown below. Use a midpoint sum with 3 sub-intervals to estimate the area between the graph of h and the x-axis on [0, 3].

Dividing [0, 3] into 3 equal sub-intervals gets us sub-intervals [0, 1], [1, 2], and [2, 3], with midpoints 0.5, 1.5, and 2.5. The rectangles we get look like this:

The width of each sub-interval is 1. Adding up the areas of the rectangles we get

1(1) + 3(1) + 4(1) = 8.